General Properties of Waves

Objectives

The objectives cover understanding wave motion through examples like ropes, springs, and ripple tanks, including the use of the term wavefront.
Understanding that waves transfer energy without transferring matter.
Defining and using the terms speed, frequency, wavelength, period, and amplitude, including graphical representation.
Applying the relationship between speed, frequency, and wavelength to solve problems: $v = f \lambda$
Comparing transverse and longitudinal waves with examples.
Understanding that sound is produced by vibrating sources and requires a medium for transmission.
Describing the longitudinal nature of sound waves in terms of compressions and rarefactions.
Relating loudness to amplitude and pitch to frequency.
Describing how sound reflection produces echoes and their use in measuring distances.
Describing and explaining the uses of ultrasound, such as sonar and medical scanning.

10.1 Waves

A wave is a disturbance that transfers energy from one place to another without transferring matter.
A wave is made up of periodic motion.
Periodic motion is motion repeated at regular intervals.
The source of a wave is a vibration or an oscillation.

Examples of Wave Motion

Wave motion through a rope

When one end of a rope is fixed to a wall, oscillating the free end generates rope waves that move towards the wall.
Kinetic energy is transferred from the hand to the wall through the rope particles (medium) via rope waves.
As the rope waves pass through the rope particles, these particles gain kinetic energy and move up and down.

Wave motion through water in a ripple tank

In a ripple tank, an oscillating spherical dipper generates water waves that move outwards.
Kinetic energy is transferred from the dipper through the water particles (medium) via water waves.
As the water waves pass through the water particles, the water particles gain kinetic energy and move up and down.

Wavefront

A wavefront is an imaginary line joining all adjacent points on a wave that are in the same phase.
The distance between two consecutive wavefronts is equivalent to one wavelength.

Transverse and Longitudinal Waves

There are two types of wave motion: transverse and longitudinal waves.

Transverse Waves

Waves travel perpendicular to the direction of vibration.
Vibration creates regions of crests (maximum displacement) and troughs (minimum displacement).
Example: Light waves.

Longitudinal Waves

Waves travel parallel to the direction of vibration.
Vibration creates regions of compressions (particles are closest together) and rarefactions (particles are furthest apart).
Example: Sound waves.

10.2 Properties of Wave Motion

Displacement-Distance Graph

A displacement-distance graph of a transverse wave can be used to determine the amplitude and wavelength of the wave.

Crests

The highest points of a transverse wave.

Troughs

The lowest points of a transverse wave.

Amplitude

The maximum magnitude of displacement from the rest position.
Measured from the rest position to a crest or a trough.
SI unit: metre (m).

Wavelength ( $\lambda$ )

The distance between two successive crests or troughs, or the shortest distance between any two points which are in phase.
SI unit: metre (m).

Displacement-Time Graph

A displacement-time graph can be used to determine the amplitude, period, and frequency of the wave.

Period (T)

The time taken to produce a complete wave.
SI unit: second (s).

Frequency (f)

The number of complete waves produced per second.
SI unit: hertz (Hz).
The relationship between period and frequency: $f = \frac{1}{T}$

Wave Speed (v)

The distance travelled by a wave in one second.
$v = \frac{\lambda}{T} = f \lambda$
Where $\lambda$ is the wavelength and T is the period.
SI unit: metre per second (m/s).

10.3 Production and Transmission of Sound

Sound wave is a form of energy transferred as a longitudinal wave through a medium, created by a vibrating source.
An object vibrating in a medium displaces particles, producing sound waves that propagate through compressions and rarefactions.
Compressions are regions of higher pressure due to particles being closer together.
Rarefactions are regions of lower pressure due to particles being spread further apart.

Example of sound waves produced by vibrating prongs:

When prongs move outwards, they create a compression.
When prongs move inwards, they create a rarefaction.
The wavelength ($\lambda$) of a sound wave is the distance between the centers of two successive regions of compressions or rarefactions.
The wave speed equation, $v = f \lambda$ , is applicable for sound waves.
Sound waves require a medium (solids, liquids, or gases) to travel.

10.4 Pitch and Loudness

Pitch

The pitch of a sound wave is related to its frequency. Higher frequency means higher pitch.

Loudness

The loudness of a sound wave is related to its amplitude. Larger amplitude means louder sound.

10.5 Reflection of Sound

Echo

An echo is the repetition of sound due to reflection off a surface.
Echoes measure distances and detect object locations by measuring the time interval between sending a sound pulse and receiving the echo.
$\text{Speed of sound in medium} = \frac{\text{Total distance travelled by sound}}{\text{Time interval}}$
Note: Object distance from the sound generator is half the total distance travelled by the sound.

10.6 Ultrasound

Ultrasound is sound with frequencies above the human range of audibility (above 20,000 Hz).

Applications of Ultrasound

Locating fishes or underwater objects (Sonar).
Prenatal scans to check foetus development.
Medical imaging of soft tissues.

Worked Example 1

In a ripple tank, a dipper moves up and down at point A. The diagram shows the position of the wave crests after 6 s.

(a) Calculate the period of the wave.

(b) Calculate the speed of the wave.

Solution

(a) Since there are 4 wave crests, 4 periods have passed.

Period of the wave, $T = \frac{6}{4} = 1.5 s$

(b) Wavelength of the wave, $\lambda = 5 cm = 0.05 m$

Speed of the wave, $v = \frac{\lambda}{T} = \frac{0.05}{1.5} = \frac{1}{30} = 0.0333 m/s$

Time taken for the wave to travel from point B to point C = $\frac{0.1}{\frac{1}{30}} = 3 s$

Worked Example 2

A wave travels along a rope from left to right. Fig. (a) shows how the vertical displacement of a point X on the rope from its equilibrium position varies with time.

(a) State the amplitude of the wave.

(b) Determine the frequency of the wave.

(c) Fig. (b) shows how the vertical displacement of another point Y on the rope from its equilibrium position varies with time.

Given that point Y is 2 m to the left of point X, show that one possible value for the speed of the wave is 1 m/s.

Solution

(a) Amplitude = 2.5 m

(b) $f = \frac{1}{T} = \frac{1}{4} = 0.25 Hz$

Since T = 4s, then $t_1 = \frac{T}{2}$ and the distance, d, between points X and Y in terms of the wavelength of the wave = $n + \frac{1}{2} \lambda$ , where n is an integer.

Assuming that points X and Y are $\frac{1}{2}$ a wavelength apart, i.e. n = 0, and given that d = 2 m, one possible wavelength, $\lambda = 4m$

Speed of wave, $v = \frac{\lambda}{T} = \frac{4}{4} = 1 m/s$

Worked Example 3

A man stands between two cliffs and claps his hands once.

The first two echoes are heard 1 s and 2.8 s respectively after he claps his hands. The speed of sound in air is 330 m/s. Find the horizontal distance between the two cliffs.

Solution

Let the horizontal distance between the man and cliff A be x m, and the horizontal distance between the man and cliff B be y m.

Assuming that x < y, then the first echo heard is due to sound reflecting off cliff A, and the second echo heard is due to sound reflecting off cliff B.

For the first echo,

$\text{Speed of sound in air} = \frac{\text{total distance travelled by sound}}{\text{time interval}}$

$330 = \frac{2x}{1}$

$x = 165 m$

For the second echo,

$\text{Speed of sound in air} = \frac{\text{total distance travelled by sound}}{\text{time interval}}$

$330 = \frac{2y}{2.8}$

$y = \frac{330 \times 2.8}{2} = 462 m$

Therefore, the horizontal distance between the two cliffs = x + y = 165 + 462 = 627 m

Summary

Wave

Definition: A disturbance that transfers energy from one place to another, without any matter being transferred.

Types

Transverse Wave:
- Wave travel perpendicular to direction of vibration.
- Creates regions of crests and troughs.
- E.g. Light waves.
Longitudinal Wave:
- Wave travel parallel to direction of vibration.
- Creates regions of compressions and rarefactions.
- E.g. Sound waves.

Graphs

Displacement-Distance Graph:
- Can obtain Amplitude and Wavelength ($\lambda$).
- Amplitude: The maximum displacement from the rest position.
- Wavelength ($\lambda$): The shortest distance between any two points which are in phase.
Displacement-Time Graph:
- Can obtain Period (T) and Frequency (f).
- Period (T): The time taken to produce a complete wave.
- Frequency (f): The number of complete waves produced per second.

Wave Speed

The distance travelled by a wave in one second.
$v = f\lambda$
Definition: An object vibrating in a medium.

Sound Characteristics

Amplitude is related to Loudness.
Pitch is related to Frequency.

Transmission

By the displacement of particles in the medium that forms a series of compressions and rarefactions.

Compression

Region of higher pressure than surrounding pressure due to particles being closer together.

Rarefaction

Region of lower pressure than surrounding pressure due to particles being spread further apart.

Ultrasound

Frequency range: > 20 kHz
Is used in Medical imaging and Sonar.

Echoes

Repetition of sound due to the reflection of sound off a surface.
Is applied to measure large distances and detect location of objects.
$\text{Speed of sound in medium} = \frac{\text{Total distance travelled by sound}}{\text{Time interval}}$

General Properties of Waves